Seminar & Events Bulletin: Differential Equations
چکیده
Differential Equations Thursday, September 24, 2015, 4:00pm-5:00pm 4088 East Hall Zhiwu Lin (Georgia Tech) Instability index, exponential trichotomy and invariant manifolds for Hamiltonian PDEs Consider a general linear Hamiltonian system u_t = JLu in a Hilbert space X, called the energy space. We assume that L induces a bounded and symmetric bi-linear form on X, and the energy functional has only finitely many negative dimensions n(L). There is no restriction on the anti-selfadjoint operator J, which can be unbounded and even with an infinite dimensional kernel space. Our first result is an index theorem on the linear instability of the evolution group e^tJL. More specifically, we get some relationship between n(L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. Our second result is the linear exponential trichotomy of the evolution group e^tJL. In particular, we prove the nonexistence of exponential growth in the finite co-dimensional center subspace and the optimal bounds on the algebraic growth rate there. This is applied to construct the local invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases, we can prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM and Boussinesq equations, 2D Euler equation for ideal fluids, Vlasov-Maxwell systems for collisionless plasmas etc. This is a joint work with Chongchun Zeng. http://www.math.lsa.umich.edu/seminars_events/ Page 3/9 Seminar & Events Bulletin: Differential Equations 07-01-2015 to 06-30-2016 Differential Equations Thursday, October 01, 2015, 4:00pm-5:00pm 4088 East Hall Jing-Rebecca Li (INRIA Saclay/Ecole Polytechnique) Numerical Simulation and Macroscopic Model Formulation for Diffusion Magnetic Resonance Imaging in the Brain Diffusion Magnetic Resonance Imaging (dMRI) is an imaging modality that gives a measure of water diffusion in biological tissue. As water diffusion is strongly affected by the cellular environment, dMRI has become a widely used research and clinical tool for detecting and quantifying physiological and pathological conditions in-vivo, in particular in the brain. In-vivo brain dMRI measures water diffusion (Brownian motion) during a diffusion time that is usually limited to between 10-40ms, and the dMRI signal is the average water proton magnetization over a voxel that is on the order of 8mm^3. Inside the voxel, there are many neurons, glial cells, and the extra-cellular space. Neurons have a solid cell body with attached axons and branching dendrites, while the extra-cellular space is very thin and geometrically complex. In addition, the cells are permeable to water, so water can move between the cells and the extra-cellular space. Due to such complex structure, the inverse problem of obtaining geometrical information on the brain tissue from the dMRI signal is a challenging and unsolved problem. I will discuss two aspects of the modeling and simulation problem for brain dMRI. The first is the direct simulation of the dMRI signal from a prescribed cellular-level geometrical configuration defined in a voxel. The second is the formulation of macroscopic (voxel-level) models of the dMRI signal for the purpose of inverting for model parameters from the experimental signal. I will review the main existing approaches and describe current progress as well as future challenges. In particular, for direct simulation I will cover 1) the random walkers/Monte Carlo approach, and 2) the numerical solution of Bloch-Torrey partial differential equation (PDE). For the macroscopic modeling I will describe 1) the more heuristic approach that models cell components of the brain tissue by cylinders and spheres, and 2) the more mathematical approach that applies homogenization techniques to the Bloch-Torrey PDE. http://www.math.lsa.umich.edu/seminars_events/ Page 4/9 Seminar & Events Bulletin: Differential Equations 07-01-2015 to 06-30-2016 Differential Equations Thursday, October 08, 2015, 4:00pm-5:00pm 4088 East Hall Greg Galloway (University of Miami) On the geometry and topology of initial data sets in General Relativity An initial data set in spacetime (M,g) consists of a spacelike hypersurface V, together with its induced (Riemannian) metric h and second fundamental form K. A solution to the Einstein equations for physically relevant sources influences the geometry of V via the Gauss-Codazzi equations. In this talk we present some results concerning the geometry and topology of asymptotocally flat initial data sets in spacetime. In particular, we will consider the topology of black holes in higher dimensional gravity, inspired by certain developments in string theory and issues related to black hole uniqueness. We will also discuss recent work on the geometry and topology of the region of space exterior to all black holes, which is closely connected to the notion of topological censorship. The results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. This talk is based on works with a number of collaborators: Lars Andersson, Mattias Dahl, Michael Eichmair, Dan Pollack and Rick Schoen. Differential Equations Thursday, October 15, 2015, 4:00pm-5:00pm 4088 East Hall Shuang Miao (Umich) On the motion of the free boundary of a self-gravitating incompressible fluid The motion of the free boundary of an incompressible fluid body subject to its self gravitational force can be described by a free boundary problem of the Euler-Poisson system. This problem differs from the water wave problem in that the constant gravity in water waves is replaced by the nonlinear self-gravity. In this talk, we present some recent results on the well-posedness of this problem and give a lower bound on the lifespan of smooth solutions. In particular, we show that the Taylor sign condition always holds; we prove that for smooth data of size $\epsilon$, a unique solution exists and remains smooth for time greater than or equal to $O(1/{\epsilon^2})$. This is achieved by constructing an appropriate quantity and a coordinate change, so that the new quantity in the new coordinate system satisfies an equation without quadratic nonlinearity. This is a joint work with L.Bieri, S.Shahshahani, and S.Wu. http://www.math.lsa.umich.edu/seminars_events/ Page 5/9 Seminar & Events Bulletin: Differential Equations 07-01-2015 to 06-30-2016 Differential Equations Thursday, October 29, 2015, 4:00pm-5:00pm 4088 East Hall Chunjing Xie (Shanghai Jiaotong Univ./CMSA, Harvard) Steady Euler flows past a wall or through a nozzle In this talk, we discuss the well-posedness theory for the steady Euler flows past a wall or through a nozzle. The steady Euler system is reduced to a single second order equation when the streamline of the flows have simple topological structure. Our focus is on the subsonic flows with large vorticity. Differential Equations Thursday, November 05, 2015, 4:00pm-5:00pm 4088 East Hall Zaher Hani (Georgia Tech/MSRI) Out-of-equilibrium dynamics and statistics of dispersive waves. Out-of-equilibrium behavior is the characteristic feature of the long-time dynamics of nonlinear dispersive equations on compact domain. This means that solutions typically do not exhibit any form of long-time stability near equilibrium solutions or configurations. We shall survey several aspects of this behavior both from a dynamical systems and statistical mechanics point of view. Differential Equations Thursday, November 12, 2015, 4:00pm-5:00pm 4088 East Hall Sijue Wu (Umich) On two-dimensional gravity water waves with angled crests We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-right angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem, and we show that the two-dimensional water wave problem is solvable locally in time in this framework. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with $-\frac{\partial P}{\partial \bold{n}} \ge 0$, instead of the strong Taylor stability criterion $-\frac{\partial P}{\partial \bold{n}} \ge c > 0$. This work is partially joint with Rafe Kinsey. http://www.math.lsa.umich.edu/seminars_events/ Page 6/9 Seminar & Events Bulletin: Differential Equations 07-01-2015 to 06-30-2016 Differential Equations Thursday, November 19, 2015, 4:00pm-5:00pm 4088 East Hall Zhouping Xin (Chinese University, Hongkong/CMSA, Harvard) Entropies and Uniqueness of Weak Solutions to Multi-Dimensional Compressible Euler Systems For the ideal compressible Euler systems, which are fundamental in compressible fluid-dynamics and pro-type examples of nonlinear hyperbolic systems, one of the main features is that the characteristic speeds of a wave propagation depend on the wave itself which leads to the finite-time formation of shocks in general. Thus one has to work with weak solutions globally. Yet the uniqueness of the "physical" solutions becomes a challenging issue. In the one-dimensinal case, various admissible criterion have been introduced to rule out the non-physical solutions. In particular, the physical entropy can guarantee the uniqueness of weak solutions at least in the case of small variations. However, in higher space dimensions, for some given initial data, there are infinitely many highly oscillatory solutions (wild solutions) which are bounded, measurable and satisfying the physical entropy condition. In this talk, I will review some progress on the constructions of such "wild solutions" by a method of convex integration; present some results on the structure of such "wild solutions"; and investigate the effects of lower order dissipations. Some open problems will be discussed too. Differential Equations Thursday, December 03, 2015, 4:00pm-5:00pm 4088 East Hall Jacob Bedrossian (Univ. Maryland, College Park) On the subcritical transition of the 3D Couette flow We discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions below this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical observations. The primary linear stability mechanism is an anisotropic enhanced dissipation effect which results from the mixing caused by the large mean shear; the main linear instability mechanism is a non-normal instability known as the lift-up effect. Understanding weakly nonlinear effects and devising the correct norms to estimate them form the core of the analysis we undertake. Joint work with Pierre Germain and Nader Masmoudi. http://www.math.lsa.umich.edu/seminars_events/ Page 7/9 Seminar & Events Bulletin: Differential Equations 07-01-2015 to 06-30-2016 Differential Equations Thursday, February 25, 2016, 4:00pm-5:00pm 4088 East Hall Matthew Creek (Univ. of Chicago) Global Well-Posedness Results for Generalizations of the Nonlinear Sigma Model The classical nonlinear sigma model of Gell-Mann and Levy, which describes interactions between nucleons and pions, has given rise to several generalizations. Among these are the Skyrme and Faddeev models, which are quasilinear generalizations that admit topological solitons. The global well-posedness of the equations of motion associated with these models has been studied intensely in recent years, in both the smalland large-data regimes. In this presentation, I will survey some of the current results related to these models. Then I will state and outline the proof of a large-data global well-posedness result of mine concerning the two-dimensional Skyrme model. This proof features a nonstandard technique which is not well known. Finally, I will suggest some possible future projects related to this work. Differential Equations Thursday, March 10, 2016, 4:00pm-5:00pm 4088 East Hall Liliana Borcea (Univ. of Michigan) Electromagnetic wave propagation in random media We study Maxwellâ€TMs equations in media with small random fluctuations of the electric permittivity, to obtain a detailed mathematical characterization of the statistics of the electric and magnetic fields at long distances of propagation. We introduce a novel wide-angle wave propagation regime, which is mathematically justified by scaling assumptions. In this regime, we obtain a decomposition of the waves in transverse electric and magnetic modes with random amplitudes. These amplitudes account for the cumulative scattering effects in the medium, and satisfy a system of stochastic differential equations which can be analyzed with the Markov limit theorem. The result is an explicit quantification of the randomization of the waves due to scattering, an understanding of polarization effects, and a mathematical justification of the radiative transport equations with polarization. http://www.math.lsa.umich.edu/seminars_events/ Page 8/9 Seminar & Events Bulletin: Differential Equations 07-01-2015 to 06-30-2016 Differential Equations Thursday, March 17, 2016, 4:00pm-5:00pm 4088 East Hall Percy Deift (Courant Institute, NYU) Ziwet Lecture 3: The Toda eigenvalue algorithm: Universality of fluctuations of halting times This is joint work with Tom Trogdon. The speaker will show how to prove rigorously universality of the fluctuations in the halting times described in Talk 1, in the particular case of the Toda eigenvalue algorithm. Differential Equations Thursday, March 24, 2016, 4:00pm-5:00pm 4088 East Hall Zaher Hani (Georgia Tech) TBA Differential Equations Thursday, March 31, 2016, 4:00pm-5:00pm 4088 East Hall Volker Elling (Univ. of Michigan) TBA http://www.math.lsa.umich.edu/seminars_events/ Page 9/9